By Art Duval, Contributing Editor, University of Texas at El Paso
One of the highlights of my summer was attending a research conference, Stanley@70, celebrating the 70th birthday of my Ph.D. advisor Richard Stanley. Because it was a birthday conference, many of the speakers went out of their way to say a little something about Richard Stanley, with mathematical or personal anecdotes. One talk in particular, by Lou Billera, did an especially good job giving the history and context of the study of face numbers of simplicial polytopes, in which Richard played an essential role. (The slides don’t totally convey the breadth of the talk, but at least give you some idea of the mathematical story he was telling.) I really appreciated Lou’s talk, and I know (from asking them) that other participants did too. This got me thinking that the mathematical community could do more of this sort of thing, not just at conferences, but more importantly in courses for our undergraduate majors and graduate students. In these courses, we rightfully focus on the truth of mathematical results. Let’s also spend some time sharing with our students why we care about the mathematical objects and ideas that show up.
We’ve long been blessed in mathematics to have the freedom to not worry about applications of our discipline. My favorite expression of this attitude comes from Bernd Schröder whose slides at a recent talk jokingly answered the question “Who cares?” with “Who cares who cares? It’s cool.” This was his clever way of expressing his observation that enthusiasm can override pragmatism. (It is only fair to note that Bernd advocates for applications, and that he subsequently gave more specific reasons to care about the topic of his talk.) In other words, we are free to investigate whatever looks interesting to us, and I value that freedom just about every day. But what looks interesting to us, and why?
Of course, in many settings it is the application that makes a result or topic interesting, and I don’t mean to diminish this motivation in the least. Many studies [8, 9] recommend including more applications in mathematics classes at all levels, not just for the sake of the application, or its use for students who are in (or who will be going into) science and engineering, but to help students better understand the underlying mathematics itself. Indeed, the five strands of mathematical proficiency in , including “strategic competence” (problem formulation and solving), are specifically described as “interwoven and interdependent”. But these recommendations tend to point towards the K-12 classroom, or towards applied or lower-division undergraduate courses, such as calculus. This same principle seems to me no less relevant in upper-division pure mathematics courses for our majors, and even graduate courses: You can get a better handle on an idea if you know where it came from, or where it is going.
Here are some questions for students to ask or for teachers to answer, even in pure mathematics. The answers don’t need to be long or detailed. Why did people start looking at this topic? What were the motivating examples? How did the ideas develop? How is it used in other areas of mathematics or outside mathematics? Why is this topic in this textbook, or why is this course being taught?
Sometimes a topic, theorem, or definition is just inherently interesting for purely mathematical reasons, which will have resonance for mathematics students who already appreciate abstract thinking. Some quick examples from the mathematics I’m most interested in include symmetry of structures, large matrices whose eigenvalues are integers, and large polynomials that factor linearly. But even when something is inherently interesting to experienced mathematicians, it can be worthwhile to take the time and effort to point this out to students who are just beginning their careers as mathematicians or teachers, and who may not have yet developed that same appreciation.
The needs of future mathematics teachers in this regard may be a little different than those planning to go to graduate school in mathematics. For this cohort, by far the most important context is “How will this show up in my high school (or middle school) classroom?” (See  for more detail.) Here, some of the textbooks for capstone courses for teachers, for instance [2, 10], have good ideas, which can be incorporated into other courses as well. For instance, the plethora of structures introduced in an algebra class have important examples in high school, which may be helpful for other students as well: The reals and rationals are fields, integers are a ring, and polynomials and matrices each form an algebra, etc. One textbook we’ve used  illustrates the need for all the field axioms by showing that these axioms are exactly the rules we need to solve linear equations.
Students don’t have to wait for instructors to do this for them. Students can ask the questions above, or make up new ones. Students may consult good books with historical and contextual material. Teachers can strongly influence this type of self-study by recommending references, for example the mathematics history books listed in the “What to read next” chapter of  and mathematics biographies in the “review of the literature” in , and also some of the other references listed below. Teachers can find other ideas for how to guide students to mathematics history at Reinhard Laubenbacher and David Pengelley’s excellent website for teaching with original historical sources in mathematics, which has resources for single projects or entire courses [6, 7].
All this is not to say we should stop doing what we normally do in pure mathematics classes. Of course, the careful definition-theorem-proof development of topics is the backbone of mathematics. This is what lets us be certain of our results, which is the other blessing we have working in pure mathematics. But it can also lead to students’ misconception that mathematics is created in this order: First come the definitions, which cannot be changed once they are written down, and then theorems are stated, and subsequently proved. Those of us in the business know it doesn’t usually work this way! We should let students in on the secret that the process is a lot more circular than the textbooks let on.
A nice example of this messiness that also illustrates some other ideas here is the notion of an ideal in a ring. It is usually introduced in algebra books simply with the definition, and then some basic results about it are stated and proved. But why would you want to work with this definition? To summarize greatly (see  for much more detail), ideals started with Kummer’s introduction of “ideal numbers”, generalizing integers by considering the set of multiples of one number, or, more broadly, a set of numbers. But one reason (which I explicitly share with my algebra students) we see them so much is that if you need to make a quotient ring, then the definition of an ideal gives you exactly what you need in order for this quotient to be well-defined. (This is a good exercise if you haven’t thought about it before.) Each of these extra facts about ideals reinforces points you would probably want to make anyway.
Let me finish where I started, with Lou Billera. When I wrote to him about including a reference to his talk in this post, we had a nice email conversation about these ideas. I’ll give Lou the last word, from that conversation:
When I first started teaching here, I became aware of several older professors (in engineering) whose class lectures consisted of what I called “war stories”, e.g., how they solved this or that problem for this or that company. I thought they were just wasting their time BS’ing and not “covering the material”, but the students loved it. In the end, for them, the “war stories” were probably much more useful in their professional lives as engineers than the “material” ever could be. (Besides, the “material” was in the book, and they all knew how to read.) To the extent we can get “war stories” into our own mathematical teaching, without sacrificing “the material” (too much), our students will be better off for it.
 Berlinghoff, W., & Gouvêa, F. (2004). Math through the ages: A gentle history for teachers and others. Farmington, ME: Oxton House Publishers; and Washington DC: Mathematical Association of America.
 Bremigan, E., Bremigan, R., & Lorch, J. (2011). Mathematics for secondary school teachers. Washington, DC: Mathematical Association of America.
 Conference Board of the Mathematical Sciences (2012). The mathematical education of teachers II. Providence, RI: American Mathematical Society; in cooperation with Washington, DC: Mathematical Association of America.
 Hersh, R., & John-Steiner, V. (2011). Loving and hating mathematics: Challenging the myths of mathematical life. Princeton NJ: Princeton University Press.
 Kleiner, Israel, (1996, May). The genesis of the abstract ring concept. The American Mathematical Monthly, 103(5), pp. 417-424.
 Knoebel, A., Laubenbacher, R., Lodder, J., & Pengelley, P. (2007). Mathematical masterpieces: Further chronicles by the explorers. New York, NY: Springer.
 Laubenbacher, R., & Pengelley, P. (1999). Mathematical expeditions: Chronicles by the explorers. New York, NY: Springer.
 National Research Council (2001). Adding it up: Helping children learn mathematics. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
 Schoenfeld, Alan (2007). What is mathematical proficiency and how can it be assessed? In Assessing mathematical proficiency (pp. 59-73). New York, NY: Cambridge University Press.
 Usiskin, Z., Peressini, A., Marchisotto, E., & Stanley, D. (2003). Mathematics for high school teachers: An advanced perspective. Upper Saddle River, NJ: Pearson Education.